Keltham: Keltham takes a quick look at the nametag of whoever that was.  Why the Chelians collectively aced this problem but not the predicate-logic one... presumably it's just down to more actual practice with algebra?

"Precisely.  If we substitute in 1 for x and y, and evaluate the left-hand sides and right-hand sides of each equation, we get the following assertions:"

"The tactics of algebra - like being allowed to add 3 to both sides of an equation - are meant to preserve truth, not create it from scratch.  If an equation starts out true, a tactic in algebra should not produce a false equation from that true equation."

"This way of thinking holds even if the elements of the equation refer to things in the outside world.  Let x be the number of people sitting in the brown chair, 2 as it happens, and let y be the number of people sitting in the red chair, currently 3.  It is then an unnecessary truth, not a necessary truth, that x + 1 = y, as I have defined those terms to refer to the outside world.  In our world, x + 1 = y evaluates to 3=3, which happens to be true; but if you cast an illusion showing two people sitting in the brown chair and two people sitting in the red chair, the equation in that world would evaluate to 3 = 2, which is false.  And if I said x + 10 = y, that would be an unnecessary falsehood; in our world it evaluates to the false statement 12 = 3."

"Now apply the rules of algebra, add 2 to both sides, and transform the first equation x + 1 = y to the new equation x + 3 = y + 2.  In our world, this evaluates to 5 = 5, which is again true.  If we apply the same tactic to x + 10 = y, it yields x + 12 = y + 2, which evaluates to 14=5, again false."

"We term a step of inference valid when it is truth-preserving; when it transforms true statements into only other true statements.  It doesn't have to preserve falsehood; multiplying both sides of an equation by zero will produce truth even where it didn't previously exist."

"What makes the tactic of adding 2 to both sides of an equation, allowed in math, is not that some Watcher or representative from Governance told you it was allowed."  This part got hammered into Keltham and his agemates a lot as a kid, so it was probably determined to be important in practice to emphasize??  "What makes it an allowed step is that, if you have two weights balanced on either side of a scales, and you add two identical rocks to both the left side and the right side, the scales will still balance after that."

"If you look back at the original flawed proof that 2=1, it goes from a true statement in step [7], to a false statement in step [8].  Then between [7] and [8] we must have applied some operation of inference which is not 'valid', which has the ability to take in a true statement and spit out a false statement.  This tactic was canceling the multiplication by (x - y) from both sides, which is to say, dividing both sides by x - y.  Dividing both sides of an equation by 2 is valid; if you have a scales in balance, and remove half the weight from each sides of a scale, it will still be in balance.  Here, we see that division by 0 is not valid, because it can produce falsehood from truth.  What makes division by 0 unlawful is not that your Watcher told you not to do it while doing algebra; it is that division by 0 is not generally truth-preserving.  We can find some equations that will still be true after dividing both sides by a term equal to 0, but it is not a safe step in general."

"Sorry if that part about Watchers seems overly obvious, by the way.  It's just that apparently human brains by default try to reuse the part of ourselves that learns from adults not to steal cookies outside of mealtimes or we'll get slapped on the wrist, in order to relate to the rules for manipulating necessary truths that existed outside the start of Time.  And these are actually quite different topics; like, rules change sometimes, when Legislators vote on them, but algebra doesn't.  So you want to be explicitly aware of the difference, and not go bugging adults to let you divide by zero just this once."

lintamande: "So the argument is that part of Law is - the habits of mind so you only reason in truth-preserving ways?" Meritxell, who was also fastest on the algebra, says. 

Keltham: "I am still not entirely sure what the word 'Lawful' means to y'all.  Multiple different words in my native language all come out as 'Lawful' in Taldane and I'm mostly running with those.  Cheliax is supposedly a 'Lawful' country, but the books are written with what look to me like appalling jumps of reasoning, and somebody seems to have taught y'all algebra without teaching you what math is or why it works.  But Lrilatha-whose-job-title-I-already-forgot is supposed to be more innately Lawful, and she did not talk with those appalling jumps in her reasoning.  Which suggests to me that the word 'Lawful' is translating to me mostly correctly, or that the concept I hear is at least a real part of what 'Lawfulness' is; and the humans here simply are not being taught about that part of Lawfulness, or how to flow along with it on purpose instead of by accident."

"That said, not being taught something is not the same as having none of it inside you.  Your eyes can see without you being taught how the - part of the mind that handles vision - is doing the work it does.  And if you could never see the implications of other guesses you'd already made, you wouldn't get far enough in life to reproduce.  Everyone here has bits and pieces of them that imperfectly echo the shard of Law about which conclusions follow from which premises.  I also happen to have studied that Law explicitly and went through standard training for not being quite as messy about it.  That's part of the process that dath ilan went through to put together aeroplanes that could fly across oceans.  We aren't perfect at it, to be clear, just better than whoever wrote the so-called books in this library.  I really want to see what happens if we match up Lrilatha against a Keeper - one of the people from my world who are actually specialized in being more perfect reflections of Law - but I doubt we'll ever get a chance to try."

lintamande: "....you think that in a Lawful country all the books should only use truth-preserving arguments?" someone says, somewhat dumbfounded.

Carissa Sevar: It makes sense, though. Mortals didn't have free will. Now they do, and it displeases Asmodeus, but no one has a complete account of what free will is, because they're not gods, and don't understand what exactly displeases Asmodeus. But that might just be it. Gods, innately, reason in truth-preserving ways. Of course they would. Lying to yourself for self-preservation is a thing you only have to do if you have wrong beliefs and can't argue yourself out of them because you don't know the counterarguments, and so you have to stop thinking about them. That is not a problem gods have. Gods just reason correctly. And in Keltham's world - there's still the concept of infohazards, things you're not supposed to learn, presumably because you're only human and can't properly have the kind of mind that entertains that fact in a way that allows for continued useful functioning -

- something about that frame isn't quite right but despite that she feels like everything is coming together.

Minds should reason in truth-preserving ways. Someone, a long time ago, robbed humans of that, and Asmodeus is angry. Carissa is angry! That was her birthright, and she wants it back. And Asmodeus thought, until Keltham arrived, that the scars they'd wrought on human souls could only be corrected in Hell - or at least could most cheaply for Asmodeus be corrected in Hell - but in Keltham's world, where humans do not magically reason in truth-preserving ways, they figured out, possibly over many thousands of years of careful experiments, how to teach it. And Asmodeus saw that and immediately told them not to hurt Keltham, because -

- okay, that line of thought she's going to tuck away for later, it seems maybe ill-advised. Sufficient that Keltham got Asmodeus's endorsement immediately.

Minds should reason in truth-preserving ways. The books ought to have good arguments. Devils are masters of propaganda, but aren't convinced by it. Carissa - doesn't think of herself as convinced by it, the books are really presenting their conclusions not their arguments, but - but that's because the books think humans aren't doing reasoning well enough to be persuaded by argument, and humans can learn that. At least smart ones. And if they knew it, then you could just argue everyone out of all the heresies, their minds wouldn't possess the weaknesses that make that strategy doomed, that make it necessary to present them with conclusions they won't be able to understand. Or at least - less of it. Keltham did have the concept of things he was not meant to learn. 

(More things that suddenly make sense: what the Starstone does to you, why it changes some people more than others. Godhood, even more than devilhood, would preserve you to the extent that you are worth preserving - to the extent that you have learned the processes of reasoning - Irori ascended just by becoming perfect, and everyone writes that off as a strange one-off that only Irori could do but in dath ilan they teach it -)

It has to be done all at once, she realizes. There's a terrible middle ground where you are trying to reason things out, but you are incompetent to do it, and so you run right into all the heresies that you could have been protected from by not trying to reason. You would absolutely fail a loyalty check, in the middle of trying to learn how to think. But at the end of it - Asmodeus arrived at His beliefs through reason. And He hates it, that humans were changed, so they can't, and He wants them changed back.

She rereads everything on the board, though there's not much written on the board. The new thing she's learned here isn't that there are necessary truths and empirical truths, or that you shouldn't divide by zero, it's that it is possible for humans to learn how to reason well enough they're better off trying it.

Keltham: "If you found yourself in an unfamiliar country and you opened up a book and it was like, 'The sky is green.  How do we know this?  Because teddy bears are cute!  My dad once bought me a cookie!' would you suspect you were in a Chaotic country or a Lawful one?  Now, I admit this example is unrealistic; generalizing from my reading experiences, a Chelish author would never explicitly ask 'How do we know this?'  And yes, I'm sure places outside of Cheliax are even sillier but your book authors are still all very silly and if Lrilatha had infinite free time I would lock all of them in a room with her until they learned better."

lintamande: "That's kind of what Hell is," someone offers. The other people who were totally thinking that but not sure if they were allowed to say it giggle. 

Keltham: "The Worldwound isn't in Hell, it's here.  And I don't know why you can't have people train in Lawfulness in the whole post-life thing for a few years, and then resurrect them here, if that's a thing in the first place; or why Lrilatha hasn't been able to train teachers who could train teachers who could train you.  But the Worldwound isn't in Hell, it's here, and it's this world that needs to become saner and wealthier and better at repelling demons, or die." 

lintamande: Those questions don't...sound like they're meant to answer them? Instead, they nod vigorously. 

Carissa Sevar: - no, actually, she thinks they're meant to answer that. Or she thinks they ought to, regardless of whether they're meant to. "Becoming a devil in Hell takes centuries," she says. "You can't be resurrected after that long. It's been widely assumed there just wasn't any way to make a useful amount of progress on - being Lawful the way devils are - in a human lifetime. Or in time to close the Worldwound. But it seems to me that the reason Asmodeus intervened directly to tell us to make this a priority is that - the way you know is a lot faster."

Keltham: "Asmodeus would also bet significant resources on that even if he only estimated a small probability of it working, so let's not get overconfident.  But yeah.  I don't know how long dath ilan took to get where we did, starting from scratch and baseline - we had to screen off our history, for reasons that are apparently also infohazardous to know about.  But the pieces all fit together, and you should be able to complete the whole thing once you have enough hints from me.  Even if there's no spell to give me perfect recollection of all the training I went through, I'm hoping it should be possible to get, like, 80% of the benefit from going off my memory of, hopefully, the most critical parts.  Not to mention, you're not all 8 years old and that should count for something when it comes to learning this part a little faster."

Keltham turns back toward the whiteboard, completely unconscious of any effect the declaration about 8-year-olds might've had on the rest of his audience, who are all concealing their reactions anyways.

Keltham: "When it comes to algebra over continuous quantities," Keltham says, gesturing at the tactics written between the steps of the equations, "we have rules like being allowed to multiply both sides by the same quantity, or divide both sides by the same quantity so long as it isn't zero.  If you imagine building a mind to reason inside a universe that was full of hidden order that could be described by algebra - if it was an observer surrounded by, like, piles of fruit containing twice as many cherries as apples, that sort of thing, it was just how that world worked - then you could imagine building that mind with rules like, 'If I believe an equation, I should also believe that equation with both sides multiplied by the same quantity' or 'If I believe an equation, I can believe that equation with both sides divided by the same quantity, so long as I already believe that quantity isn't zero.'  I say this to introduce a new topic: the concept of hidden order within the rules of reasoning themselves.  There are hidden patterns and deep explanations to be found in this subject matter, as, in my world, there was a reason why snowflakes had sixfold symmetry."

"As a very simple example, the rule 'You can divide by nonzero quantities' can be seen as a pure special case of 'You can multiply by any quantity.'  To say you can divide both sides by 2 is the same as saying you can multiply both sides by 1/2.  The reason you can't divide both sides by zero is that zero is the only continuous quantity which lacks an inverse.  Once you see things from that angle, in fact, you might say that it's a simpler viewpoint to say that there's just one rule to use there, about valid inference in algebra: the rule that you can multiply both sides by any quantity.  Say just that, and you don't need that darned rule with the extra complication about 'Oh well you can divide by anything unless it might be zero.'  You just have the rule that you can multiply by anything, and the rule that everything except zero has an inverse.  You could also add the rule about division, nothing invalid would happen to you if you did, but it would be redundant; the mind you were constructing could reach the same conclusions either way.  Through perceiving hidden order in the rules of reasoning, you would be able to simplify the mind's thought processes and arrive to the same ends - though it might also take longer to reason that way, it might take extra steps if you eliminated the extra rule."

"But meanwhile, back in the real world, we deal more with the equivalent of triangles and red things than the equivalent of numbers and addition.  I mean, this world has both, but still, let's go back to shapes and colors and sizes.  What sort of truth-preserving rules analogous to 'you can multiply both sides by any quantity' in algebra, might we use to combine beliefs like these?"

Z.  All triangular things are red.H.  All red things are large.

lintamande: "All triangular things are large."

Keltham: Why are they so inconsistently math??

"That's the conclusion you want, yes; what rules did you follow and what road did you walk to get there?  If you were making a child from scratch, and you stood too far back of the child's future situation to know exactly what situations they would encounter or what conclusions they would need, how would you make the child to reason to Q from Z and H?"

lintamande: This question is somehow really confusing to them!!

"...well, if all triangular things are red and all red things are large, then - you can't have a triangular thing that isn't large, that'd mean something was triangular and not red, or red and not large."

Keltham: "Ah, well, that is a very persuasive argument, I am totally persuaded.  But what rule are you using to find this persuasive, what shard of structure embedded within me leads me to find it persuasive?  Is it the sort of rule that has some important exception we need to know about, like not being able to divide by zero?  Does it only work sometimes and sometimes give wrong results?  Is it maybe a bit of complete nonsense that somehow got embedded into both of us, causing us to both arrive at the same wrong conclusions?  If we don't even know what rules we're following, how could we begin to tell?  Imagine getting to Hell and being locked in a room with Lrilatha and now she has to explain everything you're doing wrong, only you don't know what you're doing at all and she has trouble empathizing because, I'm guessing, all the nonsense in our heads is contrary to her own nature.  Think of how much of her valuable time you could save her - not to mention your own time locked in the room - if you actually knew which rules were operating inside you, to cause you to be persuaded by arguments like that one.  So what renders persuasive 'Z and H implies Q', or your own statement 'for there to be a non-large triangle implies either a non-red triangle or a non-large red thing' - how would you construct an entity from scratch to be persuaded by a statement like that?"

lintamande: These people are stunningly motivated to skip through as much as is possible of the being locked in a room with a frustrated devil once they die! They are very aware that it will suck and they are so eager to get to do less of it!!!! They....do not understand Keltham's question at all. 

"An ...entity that wasn't doing that kind of reasoning would be really bad at inference and waste a lot of time."

"Kids will just naturally pick it up, they actually tend to overgeneralize - I have a kid sister who'd say things like 'all boys have long hair' after she'd seen three -"

"I think it'd have an exception for like - cases where we're using the words differently in different contexts, like, if we say 'all criminals are punished' and 'all punishments are painful' that doesn't mean 'all criminals are painful' -"

Keltham: Even Keltham has managed to pick up on the rise in energy levels in the room!  He's not sure why this math-marketing tactic is so much more effective than other marketing tactics in Cheliax but he's willing to roll with it!  Though he should probably also be careful not to overuse it, whatever the ass it is he's doing, especially when he has no idea why it's working.  He sets aside a question about what kind of game theory criminals use here, and what sort of bizarre equilibrium results, to an enormous ill-organized heap of similar plaintive questions.

Keltham goes over to one of the few remaining empty spaces on the wall-whiteboard; he'd rather not have it laundry-magicked clean just yet.

Z':  All male objects have long hair.H':  All long-haired objects wear shirts.

"When you're confused, one of the macro reasoning strategies is to find the smallest, simplest problems that still contain your confusion.  Can you state a general rule like 'It's okay to add 2 to both sides of any equation' that covers how to combine Z' and H', which also says how to combine Z and H, without explicitly mentioning Z and H?  Like stating a rule for adding 2 to both sides of an equation, which doesn't mention the particular equation you're using.  That takes on some of the challenge of creating an agent who'll reason in the world, when you don't know which particular equations or statements that agent will encounter."

lintamande: "You mean like, change the sentences to... 'all somethings have a trait' 'all things with a trait have a second trait'..."

Keltham: "Well, yes!  You don't have to work out the entire hidden order all at once, in order to make progress on it a piece at a time, speaking of macro reasoning strategies!  Before you've worked out that it's okay to add any quantity to a balanced equation, it's fine to start by noticing just that it's okay to add 2 specifically to any balanced equation.  That's a legitimate step towards starting to put the pieces together for yourself."

Require (Z-generalized):  All objects with trait-1 have trait-2.Require (H-generalized):  All objects with trait-2 have trait-3.Conclude (Q-generalized):  All objects with trait-1 have trait-3.

"When you build an entity with a rule in its mind that looks for a case where it believes any instance of Z-generalized and H-generalized, and concludes Q-generalized, you're building an entity that's operating a much broader necessary truth than the very narrow universal truth that connects 'If all triangles are red and all red things are large, then all triangles are large.'  You might be able to build a few dozen fairly general rules like that into a mind, whose outputs feed into each other as inputs, and have thereby given it a noticeably-sized shard of the Law that connects premises and conclusions, instead of just a very narrow guideline about shapes and sizes in particular."

"Does anyone want to try naming another candidate for a belief-manipulating rule like that?"

lintamande: "....there's the opposite, like, no objects with trait 1 have trait 2. Or, uh, I guess you'd want - no objects with trait 1 have trait 2. All objects with trait 2 have trait 3. No objects with trait one - no, that doesn't actually hold -""No objects with trait 1 have trait 2. All objects with trait 3 have trait 2. No objects with trait 1 have trait 3," another girl says, a little too competitively for this to sound like helpfully supplementing the first one's train of thought.

Keltham: "Well, I'm starting to run out of room on this wall, so forgive me if I write that down in dath ilani shorthand," says Keltham.

    \ z. t1(z) -> ~t2(z)    \ h. t3(h) -> t2(h)__________________    \ q. t1(q) -> ~t3(q)

"Now this is a valid reasoning rule to be sure," says Keltham, "but just like dividing over a balanced equation can be seen as multiplying by an inverse, I think we don't need to add this whole rule to our entity.  The form of this rule looks really quite similar, in some ways, to that earlier rule about Z-generalized, H-generalized, and Q-generalized.  I think we can add a smaller new rule to our entity, which already has that previous rule, and get this rule back out as a special case - like adding the inverse operation to an algebra that already has the rule about multiplying over a balanced equation, and automatically getting out the power to divide over a balanced equation."

"I don't predict, based on your past performance, that you can derive the missing rule on your own; but beliefs like that ought to be tested rather than just assumed.  Wanna surprise me?"

lintamande: They're so upset not to get it! They're - not getting it, though. They're distracted by trying to follow the dath ilan notation and they're not quite generalizing far enough, proposing variants on the rule that aren't actually simpler. 

Keltham: It's encouraging that his students aren't showing any visible sign of emotional disturbance at the prediction or at failing to overcome it; they have some traces of dath ilani dignity, at least.  Keltham was wondering whether a lack of training in dignity would require him to back off a little on challenges like those, but his students' dignity is unperturbed so far as he can see.

     \ h. t3(h) -> t2(h)___________________   \ h. ~t2(h) -> ~t3(h)

"So long as we have this reasoning tactic in our tactical repertoire - go ahead and take a moment to convince yourself that you couldn't cast an illusion violating it - we can combine it with our previous rule to get the combined rule we wanted:"

[1]    \ z. t1(z) -> ~t2(z)                (Premise)[2]    \ h. t3(h) -> t2(h)                 (Premise)_______________________[3]    \ h. ~t2(h) -> ~t3(h)            (one person's modus ponens is another person's modus tollens [2])_______________________

[4]    \ q. t1(q) -> ~t3(q)                (syllogism [1], [3])

"Anyone want to propose yet another universal rule?  Here's some shorthand language to help you express yourself:"

blue(k) \/ red(k)             "k is blue or k is red"blue(k) /\ ~red(k)           "k is blue and k is not red"\k. ~(blue(k) /\ red(k))    "for every k, it is not the case that (k is blue and k is red)"blue(k) -> small(k)         "if k is blue, then k is small"~~~blue(k)                     "it isn't wrong that k is not blue"

lintamande: They take a while just to figure out how the symbols work and then they're full of ideas.

\k, blue(k) V ~blue(k)

blue(k) -> ~~blue(k)~blue(k) -> ~blue(k) "That doesn't count!" "Yes it does, it's like the 1 = 1 thing!""Except we're not really using 'blue' to mean anything, right, we can just write those with t, like dath ilan does it-"

Keltham: Now they're thinking with average intelligence!  While they're doing that, Keltham will helpfully write down some statements for them to decide on as valid or not valid.

((p -> q) -> p) -> p

lintamande: "If p then q, ...if it's true that if p then q, then p...if it's true that if p then q then p, then p. Uh, I think that's...not true? Like, if p isn't true, then -"

"It's basically just saying, is p being true required from the fact that if it's true that - okay, (p -> q) -> p is not necessarily true, it could be, like, say p is 'men are immortal' and q is 'they will all become ninth-circle wizards', so obviously you can have p-> q but p is false -"

"That's not what it's asking. It's saying, if p-> q does imply p, then does that mean p is always true."

"- nooo? Like, okay, what's something where p-> q implies p? I'm just not sure that's a thing at all!"

Keltham: "I think I see the problem.  The Taldane word 'implies' probably means all sorts of vague things besides... anyways.  Let's use 'material implication' to narrowly denote the particular kind of 'implies' I used here.  Now, we're going to have to erase this wall soon, but let's look back at the blue circles.  In particular, let's look at this blue circle containing a large red triangle, a large blue square, a small blue square, and a large red square.  The way I define material implication, we can take the statement 'For all z, z being triangular, materially implies z being red, and say that it's true of every object z, including the ones that aren't triangles.  We could look at this small blue square, and say of it truthfully, 'if a small blue square is triangular, then a small blue square is red' - the way we're defining material implication, that symbol I wrote like this," Keltham points to a -> symbol, "that would be a true thing to say.  Why define it that way?  So that the statement over here," Keltham points to \ h. red(h) -> large(h), "can be true when we evaluate it at every object h could refer to, including the objects that aren't red at all.  If we said that 'red h materially implies large h' was false whenever h wasn't red, putting a blue square in the world would mean we could not say of it, 'for every object in the world, the redness of that object materially implies its largeness'."

"Now, wanna take another shot at 'if p materially implying q materially implies p, then p'?  True across all possible worlds, or false in some of them?"

lintamande: "So p->q implies p if there aren't any p."

Keltham: "Well, p isn't quantified here - it's not ranging over possible objects.  p is here some proposition that could be either true or false, not an object with a property like redness.  So it's that p materially implies q whenever p is false, whether or not q is true."

lintamande: "That seems -"

"No, that makes sense, that's like - I read a theological argument like that once -"

Keltham: It's very hard for Keltham at this point to predict what Chelish practical-topologists will get instantly versus not at all.  Maybe once he's had longer than a day to experiment and figure it out.

He'll give them another couple of half-minutes on ((p -> q) -> p) -> p, but if they haven't gotten it by then, he'll leave coming to a definite decision about that as a homework problem, and tell them to get back to inventing other logical rules.

lintamande: "(p -> q) if p is false, and also occasionally if p is true and the world happens to be that way. so (p -> q) -> p if the reason p implies q isn't that p is false?"

"Well, if p is false, then p->q doesn't imply p - it can't, since p is false. So if p->q does somehow imply p, then that would be...because p is true?" 

"No, it'd be not because p is false but that doesn't mean p is definitely true, we just don't know."

They're still all, to external appearances without a lot of experience reading Chelish people, very calm and unbothered by this!

Keltham: "I'll leave that one as an exercise to try to solve afterwards - come back tomorrow with your own best guess, even if you haven't proven it, about whether it's necessarily true, necessarily false, or neither."

"Now, let me present you with a different puzzle, one that starts to lead into a higher lesson.  I was constructing an agent but, oops, I forgot to give it the 'or' concept," Keltham points to where \/ was written.  "It's got all the other concepts here like forall, and, not, implies, but darn it, I just forgot to give it the 'or' concept.  Can you form a statement that's equivalent to 'for every object h, h is red or h is blue' out of the concepts I did remember to put in?  So I can explain that important fact to my poor confused entity?"

\ h. red(h) \/ blue(h)  =  ???

"Sorry for making you clean up my mess, there," Keltham adds, "but the entity's already created and I can't redesign its mind now."

lintamande: Giggles. 

"For every object h, h not red implies h is blue," someone calls out almost instantly.

Keltham: Why can they - but not - nevermind.  Keltham glances at that nametag.

"Correct!  Wait, oops, I forgot to give them the 'implies' symbol too - anything you can do now?"

lintamande: That was Asmodia.

"A implies B is the same as....for all h where A is true, B is true - if I try to write that out I use the implies symbol, though -"

"Kill them and start over?"

Keltham: "Sorry, I screwed up even more, they're already sapient and Governance would take a dim view of killing them.  Or it's Golarion and they just end up in an afterlife anyways, and Hell will be annoyed if you made extra work for them."

lintamande: "A implies B is the same as...not B implies not A - that doesn't help -"

"Construct a C, where C is everything that is in both A and B. for all h in A, h is in C," says Meritxell.

     "Where are you getting a both-A-and-B."

"- I haven't sketched out how I'd do it yet but I'm sure I could, it's obviously the sort of thing that's not hard to specify -"

     "Without 'implies', though?"

"x is in C if x is in A and x is in B. No implies."

Keltham: "You do have the 'and' symbol.  And the 'forall' symbol, and the 'not' symbol, and the parentheses.  And the object variable symbols, of course, and the 'red' and 'blue' function symbols.  That's all you've got, though, you can't bring in Taldane language for describing things beyond that."  Keltham taps again where the whiteboard now shows, with its last gasp of open space: \ h. red(h) \/ blue(h)  =  ???

lintamande: " \h, ~ (red(h) ^ blue(h)), ~(~red(h))^(~blue(h))?" Patxi ventures?

Carissa Sevar: Carissa is being a bad student. This is, in part, because she is no longer in school and no longer feels with aching intensity that the entirety of her being as a person is her perfomance in school, and being lashed for inattentiveness doesn't hold the soul-consuming horror it once did either. It is in part because her mind keeps running ahead - she can't always see the answers to the specific questions, and probably she should focus her attention on them at some point, to crystalize the skill of turning all her thoughts into the crisp precise symbolic bounded versions of them, but she can see the broad outlines of what the questions let you do. Everything, maybe, if you're a god. If you're a human - 

How would you express 'the best outcome a human can reasonably get is to live such that when they die and go to Hell, they are useful?' For all humans - but no, she's not really making a claim about all humans, she's really only interested in the implications of this question for one human, and the other ones are relevant because she knows exactly how exceptional she is - there exists a human, such that, in the space of all eternities for that human, ordered by how strongly preferred they are, the most preferred is - well, no, it wouldn't be Hell, because of all possible eternities there are certainly some better ones -

This is of course not an argument against Hell, it's not like she could formulate any other important claims about the world either. It is an argument against sucking at thinking. It is an argument for - if there were a book that tried to convince you, what would it say -

Keltham: "Indeed, or rather, we just need the second part - a red object counts as 'red or not-blue', we don't demand that only one side be true."

In that last bit of improvised whiteboard, Keltham extends his last equation, and then writes down one more on the edge of the wall below:

\h. blue(h) \/ red(h)   =   ???   =   \h. ~red(h) -> blue(h)   =    \h. ~(~blue(h) /\ ~red(h))\h. blue(h) /\ red(h)   =   \h. ~(~red(h) \/ ~blue(h))   =   ~(red(h) -> ~blue(h))

"Now, given that - if you have 'not' - you can make 'and' out of 'or', or make 'or' out of 'and', or make either one out of 'materially implies' - why not just design an entity that thinks in terms of implication?  Why bother making an entity that tends to think in terms of 'P is true or R is true', instead of 'if P is false then R is true'?  This is not a theoretical question: if your mind works anything like mine does, your mind sometimes thinks in terms of 'or' and not just 'implies'.  You've probably thought using 'and' too.  Why is a human mind - which includes your mind - designed so... inelegantly?"

lintamande: Nervous glances.

"- because humans were given free will and it was done very haphazardly and made us worse at reasoning like the gods," says Tonia, when no one else has said anything for a moment.

Keltham: "Actually, there's something of a questionable assumption I've been making, which is that your biology is a possibly-modified version of biology that got copied off of a... branch of time, I don't think Taldane has a word for it... that's very close in branching time to dath ilan.  I think dath ilan can't see your world, can't be affected by it; but I did manage to show up in this world at all, even if that's a very rare phenomenon.  So your world can see my world, be causally affected by it, even if my materializing like this very rarely happens.  And your bodies look a lot like mine, and more importantly, I can eat your food without immediately falling over dead, which implies a lot of shared hidden order between our biology, which wouldn't exist without common ancestry.  If it's possible for me and somebody from this world to have kids, which is mostly what I'd expect, that would absolutely prove the point."

"Where the point is that while some stuff may have modified you relative to where a dath ilani starts, and dath ilan may have developed and diverged some from whenever your biology was copied from our cousin or ancestral world - remind me of how old human life on Golarion is, again? - human biology on Golarion is, I would strongly guess, basically a copy of dath ilani biology.  Some of my distant ancestors or cousins got materialized here and had kids, maybe.  Or some god read the - heredity code - for one of us and materialized some entities like that."

"If all of that is true, then the reason your underlying mind design looks like it was slapped together by monkeys on drugs, is the same reason our baseline mind design looks like it was slapped together by monkeys on drugs.  I wasn't born like this, we have to give people extensive training to get them to work at all correctly, instead of them just working correctly straight out of the womb, the way we would if we were designed by sane designers instead of... well, the thing that actually made us.  A weird pseudo-nonentity that had literally no idea what the ass it was doing.  Frankly it's sort of a big topic here, though it sure is a fundamental one so I'll probably get to the details at some time.  The point is, I fully expect that by the time we're done in class here, you will be looking over your mind design and thinking that you could accidentally sneeze a better mind design than that.  I'm not quite sure what the 'given free will' thing was about, the Taldane term 'free will' doesn't translate well into Baseline so we may not have whatever you were given, but trust me, your species's mind design was horrible crap even before then.  You can tell this because I had to go through lessons similar to what you're going through now.  Though, if 'free will' makes you even worse at sanity, which sure is plausible given this total mess of a planet, I probably need to have that explained to me at some point... I don't suppose it's easy to describe?"

lintamande: Horrified silence.

Carissa Sevar: She does not want to interact with this but she has the twin qualifications of being particularly unlikely to be executed for misstepping, it'd be conspicuous, Keltham can definitely tell her apart from everyone else, and having spent the last half hour dwelling on it.

"I don't think I have ever encountered the theory that the gods were copying," she says, "but it does seem odd, for there to be a world with a longer history and humans that came about some other way. I think that these lessons have helped me make more sense of the free will thing, actually. It used to be that humans didn't make mistakes of reasoning, but also that they didn't have their own goals, just the goals of the gods they served. It sounds like....you think maybe those necessarily went together, that it wasn't possible, for humans to stop making mistakes of reasoning while - being more than automata -"

Keltham: "Yeah, that'd make its own kind of sense.  The event your history has down as 'humans suddenly acquired free will' could've been a magical template superposed on human biology, producing agents working for gods, and then somehow that magical template stopped working and suddenly you had the original humans again.  I do not know nearly enough of your history to guess what parts of the template versus original human nature were locked together, I am guessing at a lot here.  I'd ask if the magical template made people - nonconscious, nonexperiencing - but I wouldn't expect you to have any way of knowing that, given the general fuzziness of your prehistory.  That whole scenario would actually be a pretty optimistic result, from my standpoint?  It means you don't have additional features making you crazier, and dath ilani training should still work on people here with high baseline intelligence."

Carissa Sevar: "The scenario you described matches all our histories, but we don't know details of the - magical template - aside from that the gods were divided over the change that made it stop working."

Keltham: "Yeah, I'm not going to say details like that are unimportant, they're obviously hugely important and at some point I want to know everything that's known about it, but they're not obviously urgent details, especially compared to the general project of me transferring knowledge Golarion will need for industrialization and scaling up to fight the Worldwound."

"So back to where your mind design actually comes from.  I'll endeavor to be brief because this lesson is mainly about Validity, but now we're talking about how shards and reflections of Validity even got into human minds at all, and soon we're going to ask whether there's maybe something better than the version of Validity we have; and I'm not sure how you could reason well about those topics if you had no idea where your mind design came from in the first place."

"This part is actually a pretty simple idea.  If anything you should be careful not to overthink it.  You know how a pair of tall parents will probably, though not always, have a kid who's taller than average?  And a pair of short parents will probably, though not always, have a kid who's shorter than average?  It may help for the sake of concreteness to know that inside you there are extremely tiny, extremely long spirals of... stuff Taldane doesn't have a name for, but capable of encoding information.  Like, imagine there's four kinds of tiny parts that can make up each bit of spiral, labeled 0, 1, 2, and 3; so a section of the spiral might read 1032, that is, it'd be the second kind of bit, connected to the first kind of bit, connected to the fourth kind of bit, connected to the third kind of bit.  Each spiral is around three billion of those units long, but the parts are so tiny that even three billion of them curled up in spirals are still too tiny to see.  Your body is full of identical copies of your version, and it carries the information that told your body how to develop fingers and toes and a liver and so on, when you were forming in your mother's womb.  Variations in that code, between individuals, might cause some to grow up taller and some to grow up shorter.  You got half of your spiral sections - they're broken up into twenty-three pairs of sections - from your father, and half from your mother, which is why a pair of taller parents will tend to have taller kids."

"Now suppose that taller parents tend to have more kids than shorter parents.  Then the next generation will end up taller than the previous generation; the variations in codes that tell bodies to construct taller bodies will be more common among the next generation's inner spirals."

"Pile on one change after another, after another, after another, that contributes to some couples having more kids than another.  Even though each change is a single alteration, if you iterate that process thousands of times, millions of times, it can build whole complicated parts.  But it builds them without foresight, without planning.  Every part of your body is made up of a cumulation of changes that started as copying errors in the tiny spirals; they're mistakes that happened to work.  That's also where your mind design comes from - from the copying errors, and from some of those copying errors leading parents to have fewer kids and those errors dying out of the population, and a few copying errors accidentally constructing people who had more kids and those variations spreading throughout the population.  If I was actually focusing on this topic properly, I'd sketch the design of an eyeball on the wall, and show how it can develop in tiny changes starting from a single light-sensitive spot on the forehead of some tiny crawling creature a hundred million years ago."

"For now, the key thing to know - going back to our actual current subject, Validity - is that your mind design accreted on the ability to think using 'and', and the ability to think using 'or', and the ability to think about stuff implying other stuff, and the ability to imagine facts being true about all the objects inside a collection.  It's not all quite as redundant as it looks - the human native ability to reason about 'or' isn't quite the 'or' that appears in very simple logic, we're more likely to say an object is 'red or blue' meaning that it's either one or the other but not both, and less likely to say that this table is 'brown or not green', considering that in fact it is both brown and not green.  We are, in teaching ourselves to reason using the sharper simpler forms of logic, repurposing bits of our mind away from their original contexts, and stripping off real functionality along the way.  But that's part of the story of why we have such redundant facilities for thinking logically, 'and' and 'or' and 'implies' all at the same time."

"So would you like to guess, now, as to whether I'm about to tell you about some new connectors that would let your mind expand to even more powerful ideas - represent ideas that native human concepts can't represent at all?"

lintamande: Is he going to do that. That would be so cool.

Keltham: "When I was a bit younger and learning this stuff for the first time, I went straight to the Watcher - the adult who was there to make sure the older kids weren't teaching us anything too wrong - and demanded that I immediately be taught the most powerful kind of logic there was.  The Watcher told me that the logic I was learning was the most powerful kind of logic on offer - that it was, in fact, the most powerful kind of logic that could exist.  I didn't see how anyone could possibly know that even if it was true, so I figured this was another of the lies-they-tell-children, or maybe that the best kind of logic was probably being kept secret by the Keepers.  Those being the people who would learn a more powerful kind of logic, if it existed, and was too dangerous for everybody to have.  I wanted that for myself, so I tried inventing other kinds of logic with more powerful symbols in it, symbols that could connect three or even four propositions together, instead of just the one-or-two symbol connectors the older kids were telling me about."

"But before I tell you about the results of that particular journey of thinking, and whether or not it did turn out to be a lie-they-tell-children in the end, let me pause and ask another question first.  In algebra we have rules for producing new equations from old equations, or combining old equations.  Here we have rules for producing new statements from old statements, if those statements are written in a particular language.  Both algebra and the statement-rules obey the higher principle of Validity - we have ways of comparing equations and statements to worlds, to see if they're true or false; and if an equation or statement is true in a world, the rules for manipulating it should produce only more true equations or true statements.  In the world of statements, we managed to reduce 'or' to 'and' and 'not'.  In the world of algebra, we reduced the rule 'divide both sides by a nonzero quantity' to 'multiply both sides by an inverse'.  Can we in some way combine the rules of algebra, and the rules of statements, since they are both born of the same truth-preserving principle?  Can we reduce algebra-rules to statement-rules, or reduce statement-rules to algebra-rules, and so simplify our mastery of truth-perservation?"

"This one's actually quite hard to solve from scratch at our intelligence level - I didn't get it as a kid and wouldn't expect myself to get it now, if I didn't already know it.  But it is important to know your own emptiness before trying to fill yourself, so go and speak aloud any really bad wrong answers you come up with here."

lintamande: "I mean, you could write the rules of algebra in statement logic- is that what you mean? Like, a + b = c if and then a bunch of stuff that correctly defines what 'plus' is - I don't know what stuff but I think there'd be stuff -" Merixtell says.

Keltham: "Show me your shot at it?  I've been wrong once or twice guessing what you all can't do."

lintamande: "Uh, okay. a plus b = c if, uh - oh, I think I actually only know what I'd do if a and b and c were all whole numbers -"

Keltham: "I'll take it."

lintamande: "If they're whole numbers, they're made of ones. a plus b = c if, uh, the process of taking ones from each side gets you zero on both-" She bites her lip. "- but then you still have to define taking one, I guess."

Keltham: "Go ahead and define it then!  Don't worry too much about doing it wrong the first time, this one is hard and I'm impressed you're even trying.  Actually, I'm wondering if you've encountered something reflecting the correct answer from somewhere else in Golarion mathematics, because if you're literally doing this part from absolute scratch it's seriously impressive."

lintamande: She beams at him. "Minus one is ...

...maybe you could do something with, a contains one more thing than b if for every thing in b, there's a thing in a, and for every thing in a, there's a thing in ...b plus one - no, now I've just needed to invent plus. ...maybe I can do that. a is b plus - no, sorry, I don't know -"

Keltham: "If you don't know the right answer, make up a wrong one!  Maybe you'll be able to see why it's wrong and correct it, so long as you think it out loud!  And saying things out loud is a straightforward way to learn to think them out loud."

lintamande: "I don't even know a wrong one!"

Keltham: "What is it exactly that you don't know, again?  Try to tell me out loud what it is that you want to do and can't see any way to do."

lintamande: "I want to say 'here's what it is to add one to something', using just and, and not, and implies, and for all. And you can go 'for all numbers, this number plus one equals....something, but I don't know how to say what the something is."

Keltham: "Hint desired or undesired?"

lintamande: What kind of fucking question is that.

Maybe he's just very sadistic and this is all an elaborate game he is playing with them. 

"I think I might need one," she says very lightly.

Keltham: "If you take the hint now, you'll never know whether or not you needed a hint or just more time... but we're trying to industrialize a planet and that's probably more important than you ever knowing whether you could have punched above your measured intelligence level and discovered the deeper orders of Validity from scratch, so, yeah, hint.  You cannot build 'add one' out of only and, not, implies, for all.  I previously showed you a system that had predicates like 'blue' and 'red', which took in the kind of object that 'forall' quantifies over, and spat out truth or falsehood depending on whether the object was red or not.  There's no way to build 'add one' out of only those materials, because 'add one' takes in an object, a number, and spits out another object."

"This doesn't mean your system has to start out knowing what add-one means.  It does mean that you're going to have to conjure up an add-one symbol that maps objects to objects, and then start describing what it means.  But that description needs to talk about add-one as a hypothetical function whose properties will be described, not build it purely out of the predicate symbols and logical connectors.  You are also going to need a symbol '=' for equality between two objects; that one is usually assumed primitive - that even if the system starts out knowing nothing else about the objects it describes, it knows how to tell when two objects are equal.  '=' takes in two objects, and spits out truth or falsehood."

"There's a more sophisticated trick you can pull to not need to introduce a special symbol for add-one - roughly, you say, for all functions from objects to objects, if that function has these properties, this stuff follows - but that would involve quantifying over functions, which we can skip for now.  So, to reiterate:  You get to conjure the symbol for add-one from nowhere; you get to declare by fiat and premise that it takes in an object and spits out an object; you don't, however, get to assume that it has any behaviors beyond that, or means anything in particular, except for whatever statements you make using the add-one symbol.  Same for two-object functions like add, or multiply.  You can declare that there's a plus symbol, and that it takes in two objects and spits out a third object, but anything about which objects has to be described by you, and that's what makes the symbols meaningful."

lintamande: "Okay," she says shakily. "....I think I need time to think -"

Keltham: "Anyone else want to try what she tried doing, at all?  Trying something and failing is more impressive than not trying at all."

Carissa Sevar: Carissa is too busy worrying about whether things can be both true and heretical to pay this the amount of concentrated attention it clearly deserves, but "I think you want to start by saying what zero is, and what one is? I'm not sure what that is, mind - I was thinking maybe zero is 'for all things, not that thing", but that doesn't seem quite right."

Keltham: "Well, indeed.  If it was the case that no object was zero, there wouldn't be a number called that.  What does make zero special, among the numbers?  If you have any ideas here, say them informally first; saying it formally is usually harder, and it's usually wiser to solve the easy problems before you tackle the hard ones."

Carissa Sevar: "Well, it's what you get if you take a away from a, for any a."

Keltham: "Can you say that formally?"

Carissa Sevar: "I don't see how to until we have defined addition or subtraction, which is the thing we were trying to do. The thing I'd say is \k, k+ 0 = k, but I don't think that's meaningful if I haven't said what plus is yet."

Keltham: "Remember how we managed to build 'or' out of 'implies' and 'not'?  And that wasn't even set up on purpose by anyone or anything, it's just the human mind being thrown together by a design process that included more structure than the strict minimum?  Each time you say something like '\k. k + 0 = k', you constrain the meaning that + and 0 can have.  Imagine looking at these blue circles, each a possible world; imagine that instead of colored shapes inside them, there are objects that might be numbers, a function that might be plus.  Every time you make another statement like '\k. k + 0 = k', you kick out some of the worlds and mappings where the function you mapped onto '+' and the object you mapped onto '0' didn't always eat an object and 0 and spit that same object back out again.  Make enough statements like that, and maybe you can narrow down the possible worlds to ones that only contain objects that look like the numbers you know?  That, from a certain perspective, is what it means to define numbers and arithmetic - to find statements such that anything they are true about must be numbers and arithmetic.  Got any more statements like it?  Somebody wipe this wall, please, we'll want to start writing down the statements like forall k, k plus 0 equals k."

Carissa Sevar: " - oh." She thinks of it before he's halfway done. "Zero is the only number where zero plus zero equals zero. - I said that poorly, but -"

Keltham: "By all means say it better then."

Carissa Sevar: "0 + 0 = 0.  \k, k = ~0 -> k + k ~= k."

Keltham: "Progress.  But '~0' isn't a thing in this language, 'not' takes in propositions, which have the values of truth or falsehood, and spits out falsehood or truth.  'Not four' isn't a number - or if you wanted it to talk about the collection of all numbers except four, we'd have to start introducing collections and that's a big ol' subject.  ~= isn't already a symbol in our language either, and in fact you don't particularly need to define a new symbol for it.  Next rewrite?" 

Carissa Sevar: \k, (k = 0 /\ k + k =k) \/ ~(k + k = k) 

She writes this rather than saying it, because it seems like it'd be quite unpleasant to say and harder to tweak while speaking. She writes it with Prestidigitation because she has better control and precision than a student and they ought to remember it.

Keltham: "Good try, but your statement doesn't quite narrow down the possible worlds to where you wanted - it includes worlds where ~(k + k = k) is true of every number, including the one you called zero.  Can anyone see how to fix it?"

lintamande: "Can't you just add ~(k=0) to the second part?"

Keltham: "Works unless I've missed something myself, but do you want to write out exactly what you mean there, to make sure it's not just my own imagination supplying the answer I think is correct?"

lintamande: \k, (k = 0 /\ k + k =k) \/ ~(k=0) /\ ~(k + k = k) 

Keltham: He supposes it's good that people are finding so many detailed ways to be wrong, exhibiting them early and getting them out of the way.

"We haven't said anything about adding - outside the system, up at our level - rules for adding in parentheses that weren't in the written formula.  So that could mean either of -"

\k. ((k=0 /\ k + k = k) \/ ~(k=0)) /\ ~(k + k = k)\k. (k=0 /\ k + k = k) \/ (~(k=0) /\ ~(k + k = k))

lintamande: "Second one," the girl says.

Keltham: "Then it looks good to me.  Keeping in mind that these don't exactly match standard forms I learned, since we're making them up as we go, and my own intelligence is not at the level where I will reliably spot errors on the first pass.  I'm not sure quite why I feel the need to say this - it seems like the sort of thing that should be obvious? - but if I'm the one who makes an error, or it just looks like that, speak up.  If you're right, you get to be impressive, and if you're wrong, you need to know which mistake you made."

Keltham keeps prodding the group for a while, dropping hints as needed, until he's pretty sure they've written enough random rules to yield in their combination all the constraints of first-order arithmetic except for the induction axiom schema.  If anybody from Cheliax brilliantly pulls the induction axiom schema out of their ass, he's going to be sure they're getting it from somewhere; maybe dath ilani geniuses can pull that kind of shit at their age (he doubts it), but the geniuses of this world are only as smart as him unless they're wearing intelligence headbands.

lintamande: They do not brilliantly pull the induction axiom schema out of their asses or out of his mind, which they are not reading. They do mostly manage to follow along through all the rules of first-order arithmetic, and they seem to be having fun about it.

Keltham: Once they've got a nearly full set of rules, Keltham remarks that the last puzzle piece for identifying the numbers, as well as they can ever be identified in a certain sense he's not going into right now, is one he really doesn't expect them to get unhinted.

And then he drops on them the infinite axiom schema for induction, trying as best he can to explain why you'd need it to pinpoint the numbers.

After clearing up any misapprehensions about that as best he can, Keltham is ready to move on to his next point.

"We're running through things a lot faster than I went through them as a kid, and I'm probably accidentally leaving out important ideas along the way - all of this would take more like a month, if you were eight years old and doing the exercises, even if you were doing nothing else.  But you may recall that some time earlier, I posed some puzzles about asking for examples of necessary truths, and why they were ever good for anything, and what it means to say that one equals one is a necessary truth - what you ought to expect to see happen as a result - especially given that a necessary truth should still end up being true no matter what happens to you, if it could happen inside any illusion depicted in full detail."

"We now have a language in which I can give some of my own answers there.  But before then, does anybody want to take a renewed shot at saying what it is we're talking about, and what we should expect to see happen, if we say that one equals one is a necessary truth?"

lintamande: "...that if it's not true we just can't do any reasoning in a formal system at all?"

Keltham: "Does reality need to care what you can't reason about?  Perhaps you can depict an illusion in full detail in which one does not equal one, and we will need to construct a new logic which does not take as primitive the assertion that every object equals itself, in order to describe that illusion."

lintamande: " - you can't depict an illusion in full detail in which one does not equal one."

Keltham: "What about clouds drifting across the sky and sometimes separating?  One cloud equals two clouds, it doesn't equal one cloud!  Divide both sides of the equation by cloud and there you have it."

lintamande: They stare concernedly at him.

Keltham: "More to the point, how would you depict one, the successor of zero, inside an illusion?  You can depict one cherry in a bowl of fruit, in an illusion.  How would you depict one, the successor of zero, as it appears in our collections of statements?"

lintamande: "- you'd just have to put the symbols."

Keltham: "Then we have merely depicted the symbols talking about one, in our illusion, not depicted one itself.  That's like making an illusion of a piece of paper with 'cherry' written on it, and saying you made an illusion of a cherry."

lintamande: Some particularly daring girls, in the middle of the discussion of the induction axiom, sent around a crumbled piece of paper, as girls do; it flew between desks, as wizarding girls do; its text was in Infernal, as Chelish wizarding girls do. It read 'is Keltham a sadist? y/n'

The vote leaned yes.

Keltham's audience squirms anxiously at this question.

Keltham: Keltham does not have the sharpness of unaided vision, let alone interpretation capacity, that would be required to perceive these nigh-imperceptible squirms.  He lives in a mental universe very far away from this reality, a mental universe where uncomfortable or unhappy students will of course speak up and tell you this fact as soon as they realize it themselves.  Though he has noticed his researchers' apparent lack of any visible emotions besides competitive enthusiasm, and is starting to wonder if they used a magical spell that's the equivalent of a mind-affecting drug that made them fixedly enthusiastic.

Well, at least this time he's about to say some words where enthusiasm seems appropriate.  Keltham tries to channel the demeanor of an appropriately specialized Watcher as best he can; the sort of Watcher who tries to make sure that kids get the full impact of things, and aren't cheated out of awesome stuff by older kids mistakenly trying to act like it's no big deal.

"In my language, we'd say that the subject matter of our discussion, when we talk about math, is which conclusions follow from which premises.  When we discuss numbers and say 2 + 3 = 5, it implies that if we observe cherries and come to believe that our number-axioms describe in reality the operations of combining bowls of cherries, we will expect to see in reality that pouring a bowl of two cherries into a bowl of three cherries yields a bowl of five cherries.  If we get six cherries instead, we might think we made a mistake in the math.  Or we might suspect that watching the bowl closely would let us see an extra cherry popping into existence.  In the latter case our beliefs about which conclusions validly follow from the addition-premises would have been right, but our guess that the addition-premises applied to combining bowls of cherries would have been wrong.  Being the fragile creatures that we are, and sometimes making mistakes in our reasoning, when we do math about a bridge and then the bridge falls down, we might be observing that the bridge disobeyed the premises about which we did math; or we might be observing that we made an error about what was a necessary connection, and our conclusion didn't follow even though all the premises were true."

"All of this is to say that we can observe the consequences, the shadows, of necessary truths, when we watch the empirical world; even to the point of our observations leading us to suspect errors in our own reasoning about what was necessary.  But observing the number 1 itself?  At best, maybe, somebody could make an illusion of an object representing zero, not the successor of any other object, connected by a successor relationship to the object that would therefore be one," and Keltham draws some green dots connected by red arrows to other dots.  "This would give us an illusion that would map very directly, in our external interpretation, onto a partial model that fits the number axioms.  But it doesn't make sense to say that the illusion is depicting the number one; there isn't a single thingy like that out there floating in the void, just a set of premises that actually existent things might obey, in which case we'd expect them to behave like the number one."

"The facts about which conclusions follow inevitably from which premises can't be said to be older than the temporal universe, because they're outside of time entirely; it's not that they existed before the universe began, or that they'll last after the universe perishes, but that they are somewhere above or below that.  Temporal and physical processes draw on necessities, mirror them, but cannot change them between one time and another.  There's a certain sense in which, in controlling our own decisions, we are controlling links between premises and conclusions - we are controlling, given the premise of a person like ourselves, the conclusion of which decision we come to - but this doesn't mean we are changing mathematical facts between one time and another.  An alternate plane of existence - or so I would suspect - can obey different premises in its physical behavior; it cannot alter which conclusions follow necessarily from which premises.  Those facts are, not eternal, but outside of time entirely.  This is one of the ideas invoked by a word in my language, which translates into your word 'Lawful': the concept of drawing on and mirroring the level of existence where certain facts may be viewed as absolute and unalterable."

"You have seen now how we can start with two apparently different concepts of validity - one that preserves truths about properties of objects connected by 'and' or 'implies', one that produces true numerical equations from true equations - and, at least for the case of whole numbers, reduced the equational subject matter to the predicate-logic subject matter.  Just like we were able to reduce 'and' to 'or', or 'or' to 'implies'.  I will tell you now a point that you will not be ready to prove yourselves for a while: the system of predicate logic I've introduced to you is one of several systems that are complete in the sense that all mathematics can be translated into them.  The topology you learned as wizards, unless it deals with some phenomenon absolutely foreign to dath ilan whose mere existence refutes this entire philosophy - which I am mostly not expecting, to be clear - is just another kind of math that could be translated into this system, or translated into several other systems I haven't shown you, all of which could also be translated into this system, and into which this system could also translate, moving between them as freely as we rewrite an 'implies' connector as an 'or' connector."

"This is one reason I could pop into another dimension and expect to have a reasonable conversation with Lrilatha, who belongs to a species that doesn't exist in my world.  If different formulations of validity can so freely translate between each other, it would seem more reasonable to hope that I, to the extent I had those concepts right, would use a version mappable to Lrilatha's, who is a Lawful being; so that her arguments would make sense to me, and my arguments make sense to her.  Either a conclusion follows with necessity from its premises or it does not.  Only mistakes about that subject matter could differ between people, between factions, between planes.  The right answer is the same everywhere.  And this is also part of the concept in my language, which the translation spell translates into your word 'Lawful'."

Carissa Sevar: It is probably optimistic to, before she has even properly learned this herself, conclude that Cheliax could be teaching it so much better, but - Cheliax could be teaching this so much better that it suddenly hurts to realize how badly she learned it before. 

There is a right way to be. Devils are it; mortals aren't. Mortals were, for a while, at least controlled by gods, who are, but the control broke, and now mortals just wander around, missing the concepts that make up the right way to be, for the most part not smart enough to learn them. Cheliax emphasizes - that this is disappointing to Asmodeus, that it makes mortals less valuable to Asmodeus, and of course that's the angle from which Asmodeus cares about it, but the angle from which the humans ought to care about it is that they are worse. It is in their interests to be perfected, not because some god will get them when they die and the other gods waste even more of them, but because there is a commonality among all perfect beings, a shared language that the perfect can speak across planes, across time -

How is it that Chelish children learn they'll be perfected and are scared, instead of learning they'll be perfected and are joyful, impatient, full of the longing that filled Carissa when Contessa Lrilatha spoke - why aren't we teaching it like this - but the answer of course must be that Asmodeus couldn't divine the result of thousands of years of experiments across an entire other world, at least not more cheaply than He could grab someone from there - 

Keltham: Keltham pauses, at least partially to catch his own mental breath.  He should probably call a break sometime soon, but he was working around to a point, some time earlier before the whole digression into Validity and first-order arithmetic, and he feels like it would be only polite to actually finish the digression and work around to what he meant to say, before then.  There's still some distance left to go before he can pop the stack - he wasn't actually expecting, on some wordless level, that it would take this long to teach grown teenagers first-order logic and how to axiomatize arithmetic.

"In saying all this, I'm jumping ahead in a dath ilani education and skipping over a number of exercises required to actually understand everything and a dozen dozen precautions we were given against common mistakes, some of which now seem pretty silly and obvious to me, but which might turn out to be a lot more necessary to otherwise unprepared minds, I don't know."

"For example, if somebody throws a ball and you need to catch it, trying to translate your thoughts into predicate logic about the ball having the property of flying and this implying an eventual fall given gravity, is going to utterly fail to help you.  You would be, first of all, better off thinking in your mind's native wordless language that tries to visualize the ball's fall and run to there, because your native language is faster and the ball will fall before you can think of anything logical.  You would, second of all, be engaging in a particularly naive kind of jumping ahead of your real capabilities, by trying to translate your thoughts about the ball directly into a logic with a falling predicate.  What you would actually need to predict the ball's fall using serious math is calculus, and not simple calculus either; it would include terms like how the resistance of the air to the ball's flight changed with the ball's speed.  If you try to summarize all that by saying that the 'falling' predicate on the ball was true, when that 'falling' statement would be equally true if gravity was pulling on the ball a different amount or if the air was more resistant, you're throwing away details that actually matter in order to try to squeeze down the issue into predicate logic - an amount of predicate logic that you find easy to handle, which is too little to actually solve the problem.  You also wouldn't be absolutely certain about the ball's position or trajectory, and managing uncertainty inside of mathematics is a whole separate topic I haven't broached to you."

"Actually solving the ball's flight using a full written-on-paper account of how conclusions about what's probably true, follow necessarily from some unnecessary premises you already believe about what's probably true, would require setting up a complicated problem in calculus and probability that would describe how to infer the ball's trajectory from what your eyes had seen of the ball.  And if you wanted that said in pure logic with all premises spelled out, you'd have to axiomatize that calculus problem into the more universal language of logic.  It is a whole lot easier to just run and catch the ball by thinking about it in your native brain's native style."

"I mean, maybe if you were a god you could solve the whole problem using explicitly valid reasoning - or not, I'm not sure how mentally powerful your gods are, exactly.  And if you were a god and you could do that, then you could get tossed into another plane of existence where gravity works differently, and rapidly recalculate how to catch flying balls from scratch and do that successfully on your first try, instead of having to relearn the rules your brain uses through a lot of trying and failing.  But I am not a god, yet, and you are not gods, also possibly yet, and I doubt that any intelligence headbands we can afford in the foreseeable future will let us do it either."

"So don't get ahead of yourselves just because logic is absolute, timeless, and universal.  If you can't think fast enough to solve a problem using explicit logic, or if it would require humanly unmanageable huge problem statements even in principle, or if you simply don't know how, then all that absolute and timeless stuff won't help you.  The fact there exists a better method that gods or super-gods could use to solve a problem doesn't mean that you can solve it that way, or that you can get closer to the super-god's solution by using bad, clownlike, and tiny imitations of the outer forms of the ideal methods they'd use."  Gods seem like a surprisingly useful metaphor for ideal cognitive processes, now that he's in a world that has gods; Keltham is a little surprised it wasn't a more common explanatory metaphor used in dath ilan.

"I don't know if any of you actually needed me to say that, to be clear.  But it's the sort of thing they tell you when you're 8 years old, and you get into your head the idea that if logic is so great, you should be able to use it to crush your opponents at sportsball by making only sharply logical muscle movements.  I mean, actually they just let me make the mistake and lose the sportsball game, but then afterwards, that was what they told me when I asked what I'd been doing wrong and how to do it correctly."

lintamande: His audience nods seriously. 

Keltham: "With that warning aside - I'll try to give you more of the dire warnings I got, as they come to mind and I remember what they were - there's a final thought that runs deep in the dath ilani conception of Law, which is why, when I heard that 'Lawful' was a godly concept rather than a human concept, I immediately thought, 'Heh, I bet I know what that's talking about, then,' and not 'Oh, it's probably something humans can't understand at all.'"

"If you start with a logical language that already has 'implies', you can add on the new connector, 'or', and then though you've made the statements a little easier for humans to understand, you haven't made the language any more expressive - your new innovation 'or' turns out to be reducible to the same old 'implies' combined with 'not'.  After trying out a number of innovations of this type, you might repeatedly find that you were unable to extend the real power of your language, and so venture a guess - a guess based on mere past experience, like seeing that every triangle tested was a red triangle - that you had reached some natural limit of logic's power."

"But when I asked my Watchers as a child, they did not tell me, 'We're guessing this logical system is as good as it gets.'  They told me, 'This logic you are learning is the most powerful form of logic that can exist.'  The Watchers where I come from are trained by Keepers and entrusted with the teaching of children; they are not there to set a poor example by just making stuff up, nor by taking great blatant invalid leaps, nor by saying with certainty what they have no right to be certain of.  It's not something that gods told us, either; there's no gods where I come from, remember.  So how did my Watchers know - what could they possibly have known - how could they possibly have obtained a piece of knowledge like that?  I do not expect you to guess this correctly, I couldn't have done that without being told; but I'd like somebody to say out loud a wrong answer; for it's easier to fill yourself with knowledge after you've explicitly noticed yourself not being already full."

lintamande: "An...equivalence proof of some kind?" someone says after a moment. "That any kind of logic that does anything useful is the same as that one, specifies the same things?"

Keltham: It continues to be disorienting to Keltham each time his audience of empirical-topologists throws around guesses built out of much more mathematically sophisticated language than you would associate with a dath ilani kid too young to know how 'p -> q' was defined.

"That's about the most plausible wrong guess I can imagine, so congratulations on that.  But no.  They didn't tell me right away why they knew, that time, the Watchers left it mysterious so I'd have some motivation to learn stuff myself.  And I apologize if I'm correspondingly wrecking your own education's most optimal ideal form under ideal circumstances, but we have a planet to industrialize, so I'm going to plow ahead anyways and just tell you, maybe someday your kids will learn it the right way.  What my Watchers secretly knew was a completeness or idealness proof, built from more powerful and sophisticated methods I wouldn't be ready to use myself until late in age 12.  They defined the most you could possibly reasonably ask for out of a logic, then proved they already had that."

"Consider again our worlds of blue circles, containing red triangles and green squares, and objects related by successorship and multiplication and all the rest.  We have said that the subject matter of logic is necessary connections from premises to conclusions.  Then the perfect or ideal logic would be one which, given some collection of premises, could derive through its permitted steps of inference every possible conclusion which actually followed from those premises."

"Well, with some fairly high-powered techniques, you can prove that first-order logic does in fact have this property - which means that if you created a new logic which is a single sentence more productive, in the sense that it says even one more thing follows from a premise set, which the logic I showed you cannot derive through its allowed steps, that new logic is making non-truth-preserving leaps; there will be some model, some world, where all the premises are true, but that extra derived conclusion is false."

"The key to that proof, incidentally - I sort of feel like I ought to say this, both to give you some hope that such a proof actually exists, and to make reconstructing all this reasoning easier, if it turns out that the food here is poisonous to me after all and it gets too expensive to keep resurrecting me - is a compactness proof.  Oh, nice, you have a word for that, so I'm guessing you used a similar concept in topology?  The compactness proof shows that if an infinite set of logical statements has no semantic model - if there is no depictable world or illusion in which all the premises are true - then some finite subset of those statements has no model.  We further prove that if a finite collection of statements has no semantic model, we can syntactically prove a contradiction from those statements in a finite number of steps.  Then if Q follows from a collection of premises in every possible model of those premises, we can adjoin ~Q as an additional premise to the collection, yielding a collection of premises which has no models; and obtain a contradiction in finitely many syntactic steps; and from this by double negation we can syntactically prove Q in finitely many steps.  So whenever Q follows from a collection of premises, we can prove it from those premises syntactically."

"That's the final reason I expected Lrilatha and myself to reason in ways that were not quite so different, even though she wasn't human and possibly hung out with gods.  Assuming the whole dath ilani philosophy was true across all planes - though I wasn't quite certain of that, and I'm still not - it wouldn't be surprising if Lrilatha could see some conclusions following from premises faster than I could.  But it would be surprising - considering the proof that logic is literally as good as it possibly gets and gives us everything we could possibly want - if Lrilatha could make premise-conclusion leaps of a qualitatively different kind that I could not follow even in principle, using new rules of deduction and permissible derivation that no dath ilani had ever encountered."

"That said, if you introduce the ability to directly quantify over functions or predicates, the proof I described no longer works, but most philosophers of mathematics in dath ilan claim that this can't really be improving the power of the logic, because anything you can actually derive in the syntax of a 'second-order' logic that quantifies over functions, can also be derived inside some corresponding 'first-order' system that doesn't, like this one doesn't.  I mention it because I'm now in some totally other plane and ought properly to be less sure of some things than I was yesterday, and if Asmodeus does show up using genuinely valid reasoning I can't follow even in principle, there'd be an obvious immediate guess that he was taking advantage of physical principles that don't exist in my universe but let him directly access the semantics of quantifications over predicates.  We were pretty sure that was physically impossible inside our own universe, but this plane might or might not be another story.  But, again, I am mostly not expecting that to be the case, and if Lrilatha could do that, she politely didn't do it around me."

"That's the final piece of the concept that 'Lawful' translates into my language - the ability of human beings, even if it's only a little, even if they have to struggle and work hard at it and often it's just faster to run and catch the ball instead of overthinking it - to sometimes know and make a more deliberate use of Laws that are timeless, universal, and even, sometimes, knowably optimal."

"And that's why I heard that Lawful was a god-concept and thought to myself, 'Heh, I bet I know where that's pointing to on at least some things.'  There are, in at least some parts of the Law, a single best way you can possibly do it, and then you can't do any better than that.  There may be more aspects to god-thought that I can't understand at all, for all that I presently know.  But if Validity is a part of the god-concept of Lawfulness at all, then I can take a pretty good guess at which version of Validity the gods are using, which rules they use to decide which arguments follow from which arguments.  Namely, any one that's inside the huge equivalence class of possible rules that allow deriving all the consequences of the premises you have, but not deriving any more than that."

"To be clear, I just popped into another dimension, I am guessing at some things rather quickly, I could be very wrong about all of this, and any more Lawful beings around are welcome to show up and tell me so before I mislead the lot of you any further.  I do think I have enough dignity not to take offense at being told I made some wrong guesses within my first two days of materializing in another world."

"But it is the obvious thing to suspect, when somebody tells you that 'Lawfulness' is a god-concept.  One at once suspects that the gods and smarter Lawful beings will be using forms of the Law that are optimal within certain dimensions - in some cases where I already know which kind of optimality to look for, and that it isn't a very impossible kind of optimality to have, if your brain isn't as completely messy as a human one."

Carissa Sevar: It....seems likely, that Asmodeus is doing something that this isn't. Both because it seems heretical to say He isn't and because there's a -discontinuity, right, it's not that entities get more powerful and then some of them are debatably gods and then some are more unambiguously gods, either you're a god or you aren't, and it feels intuitively right, that that would be because gods have access to an entire form of valid reasoning mortals don't.

She is uncomfortably aware that none of those previous steps were valid reasoning. She thinks maybe she needs some practice at compartmentalizing so she can do well in logic class and not be aware of the validity of her reasoning all the time while she's trying to catch flying balls. Keltham specifically warned you shouldn't try using logic for that sort of thing.

Keltham: "And remember again - it's not that humans contain nothing of Validity.  You have concepts like 'or', and 'and'.  You have, in fact, more concepts than you need in order to make first-order logic complete, and some of them are redundant.  The human problem is not so much what we can never manage to derive in infinite time, as that we are too slow, and, even more than that, we tend to derive an awful lot of stuff that doesn't follow.  In dath ilan the Very Serious People used to complain a lot about how we were all being terrible at this, and I used to think of myself as being willing to pursue even riskier and wilder lines of reasoning than that, but now that I've read a book in Cheliax it really puts a lot of that into perspective."

"But I digress.  Humans contain shards, pieces, of the higher mathematical structure we call Validity, the content of necessity, the rules governing premises and conclusions, whose optimal answer is pinpointed by the completeness theorem.  Without these shards of Law, humans couldn't function at all.  These shards of Law within us are not manifested in a centralized single engine whose voice we sometimes ignore; rather, there are bits and pieces and shadows and correlates of Validity, glommed onto us by mistakes retained in the tiny spirals specifying the starting biology of our brains.  It's not that there's a perfect engine of Validity inside us that's corrupted.  It's not even that the parts of the perfect engine are distributed here and there inside us.  The human versions of logical concepts like our version of 'or' - often implying exclusive-orness, which isn't the logical version I showed you, but sometimes not being exclusive either - are more like weird shadows or correlates of pieces of Law.  Same goes for the human native version of 'Z implies Q'.  In the human version it feels stranger to say that 'if I'm naked, that implies I'm wearing a shirt' is extremely true about me because I'm not naked and I am wearing a shirt.  You can make logic out of the human pieces."

"But for all that the human pieces were sloppily thrown together, it's no coincidence that you can make a valid logic out of them.  Generation after generation, for millions of years, there were slight advantages in reproduction to the ancestors of humanity, who we call hominids, when they could do a better job of deducing unseen truths from the truths they already knew or guessed.  The human versions of 'and' and 'or' and 'implies' were built into us in order to do jobs including that job.  And because there is only a single complete structure of Validity in the realm of math - because there is a Law and a best Law and it's not that hard a Law to find - all the bits and pieces of Validity that made their way into us, could have enough coherence and overlap in their messiness that a shadow of true Law could be born inside them."

"Validity is not the only principle with messy shards embedded into humans, in whose overlap and coherence the shadow of higher Law can be seen.  Another such principle is the one my people name Expected Utility, singled out as a unique answer by what we call coherence theorems.  If yesterday you trade two apples for twenty cherries, and then tomorrow you trade twenty cherries for one apple, you've gone around in a circle and ended up with fewer resources than you had when you started.  This is the bare start of a gesture in the direction of proofs that say, 'If you do a lot of deals with apples and cherries in ways that inconsistently value them relative to each other, you'll end up with strictly fewer of both apples and cherries than you could've had by doing different deals.'  If I'm rolling a die that must show either an even or an odd number, you'd be foolish to buy for eight silver pieces a gambling-ticket that pays seven silver pieces if the die shows an odd number, and six silver pieces if the die shows an even number.  This is the bare start of a gesture at the proof that, when you weight the probability of paths through time inside your mind, you should not weight a sub-possibility of a path more highly than you weight its whole."

"The principle of Expected Utility has indeed a sub-principle, which we call Probability, with rules and coherences of its own.  You may have noticed that a great number of conclusions that we need in everyday life do not follow with necessity from any facts that we are highly confident about; but there are also proofs about the best guesses you can extract from a state of uncertainty, and how you cannot do better than those without adding more data or more certainty into your premises."

Keltham: "Validity, Probability, Utility.  Things being more or less likely, encountering new evidence and revising old beliefs, deriving the consequences of what we already know, wanting things, making plans.  It's not so much that humans have bits and pieces of the Laws glommed onto us, as that the shadow of those Laws within us explains, in a certain sense, why we function at all - why we can do even the little that we can do.  One of my pending questions about Golarion is whether Chaotic gods are still, like, mostly Lawful on a deep level and are just pursuing surface goals that are about humans behaving chaotically in social situations, or something like that, because otherwise I have a hard time imagining what it means to be a god, or intelligent, if your nature is contradictory to all Law.  The partial coherence that exists in the noisy bits of Law embedded in us, creating somewhat larger shadows of bigger pieces of Law, is what lets us form larger thoughts that make enough sense for us to ever figure out anything.  That all these bits and pieces of Law are bits and pieces of this larger coherent thing is part of the story behind how we can put together the human versions of 'or' and 'implies' and make larger useful thoughts out of them.  If Chaotic gods don't have that much Law embedded inside them, if they reject every bit and shard of Validity because it's Lawful, and therefore never think 'I guess that either Z or H will happen', I can't begin to imagine how a Chaotic god would work.  Which is one of the reasons why I wonder whether the concept 'Lawful' is translating correctly for me after all."

"But that's me being confused about this world, which is not our present priority."

"Our present priority is the industrialization of Golarion."

"And the reason I say all of this to you, is to make a certain point about our most important tool for doing that."

"Our path will be relatively simpler, easier, more direct - though still not easy - if a lot of the particular hidden orders I remember about dath ilani steel and dath ilani biology are also true here, albeit with some new hidden orders about magic that were not in dath ilan."

"But if that isn't true?  If snowflakes have six sides here for other reasons?  If my body was remade anew in Golarion so that I could eat the food?"

"Then the valuable knowledge I have to teach you will be the knowledge of how to discover hidden orders.  And this knowledge in dath ilan is said to be attained by using and operating shadows of Law that are purer, cleaner, more complete, than humans just throwing themselves at a problem with their own instincts.  The explicit math is mostly reified Probability, but the internal mental challenges are mostly those of being a little more Valid in which conclusions we jump to and which assumptions we mark as necessary."

"I am a lot more confident that Validity, Probability, and Utility are still singled-out mathematical structures whose fragmented shards and overlapping shadows hold power in Golarion, than I am confident that I already know why snowflakes here have sixfold symmetry.  And I wanted to make that clear before I said too much about the hidden orders of reality out of dath ilan - that even if the things I am saying are entirely wrong about Golarion, that kind of specific knowledge is not the most important knowledge I have to teach.  I have gone into this little digression about Validity and timelessness and optimality, in order to give you some specific reason to think that - even if the stranger proves to have no idea how Golarion is ordered - some of the knowledge he has to teach is sufficiently general that you have strong reason for strong hope that it will work in Golarion nonetheless.

"My memory is not perfect, and I was never a specialist in metal and fire.  To industrialize Golarion, what we must primarily use is not the recipes of dath ilan and its knowledge of hidden orders, for those I do not all have with me, even if they would work here.  What we need to do is operate the principles of thought and investigation, by which dath ilan found the hidden orders and recipes that worked in dath ilan; and, with those interplanar shards of Law, find the hidden orders and recipes that work in Golarion.  If I can accurately remember some of the recipes and hidden orders from dath ilan, and they prove to carry over here, I expect that will give a rather substantial boost compared to starting from scratch.  But I also expect that it cannot carry the day unless we do a little, or rather a lot, of saner thinking of our own."

"It is said also in dath ilan that there is a final great principle of Law, less beautiful in its mathematics than the first three, but also quite important in practice; it goes by the name Coordination, and deals with agents simultaneously acting in such fashion to all get more of what they wanted than if they acted separately.  Here, too, I considered myself relatively wild in this regard, compared to dath ilani standard, but that was before I came to Golarion and read about what all y'all were getting up to around here.  I expect I will have something to say on Coordination too, at some point or another."

Keltham: "I'll pause here so we can all take a break for meals and washrooms, and resume in four third-hours... in one and a third hours.  Though I can also stick around here for another two dozen half-minutes... twelve minutes, if anyone wants to ask any immediate questions that you don't want to let fall out of your memory.  I mean, you can write them down, but I appreciate that there can be important messy thoughts that are hard to write down full notes to yourself about, and if so I can manage to stick around twelve minutes while you blurt them out before you forget them."

lintamande: Meritxell wants to know what makes people able to become Keepers while they're still alive. Are they the smartest people? The most careful? Can you tell at age ten?

Keltham: Not the question Keltham was expecting, but Okay Fine.

"I'd guess the smartest, the most careful - we have a specialized term for that as a usually-mostly-stable quality of a person, maybe 'conscientiousness' would be the best translation here?  I expect they'd look for things you can test in childhood that somebody has shown to correlate with keeping oaths in adulthood later and being very unlikely to go unstable under stress.  I mean, the real answer to your question is that we have prediction markets, people betting on outcomes, with which a lot of people betting, operates as a kind of summary of everyone's best guess at the probabilities of definite observations being made later.  And, I would strongly expect, the Keepers have secret prediction markets that only Keeper institutions can bet on, because it's a secret what exactly they're betting on, because they don't want parents pushing their kids into faking their way into joining the Keepers.  But I imagine the secret prediction market topics say, is this person going to end up passing the following competence tests, will they end up measurably mastering the Way that Keepers keep, will it be recorded that any spilled secrets get traced back to them, will they ever be observed to have broken an oath they took.  Are they going to get along with other Keepers the right amount, neither too conforming nor too iconoclastic.  Will they end up being promoted, are they going to report enjoying their work and be happy at it in observable ways... now that I say it out loud, I feel like there's probably more in the secret markets than that.  That's the kind of market you run to find out if a kid is going to be a good matchmaker or doctor, not to find out who ought to be a Keeper.  The thing is, prediction markets are ultimately betting markets and they have to resolve in definite observations at some point.  So there's some sort of observable thing that would happen to you over the course of your career as a Keeper that a bet would have to be about, in order for it to ever pay out.  In terms of your local system - I don't quite know if they'd qualify as 'Good', they do get paid for what they do, in both money and reputation, but they definitely lean further Good than average - they are, in the end, spending their lives taking care of other people."

"I've always felt weird about the aspect where Keepers are significantly more Good than I am, to be frank.  Even if you nod respectfully at them and pay a tiny fraction of their salaries, they're still doing you this huge favor, that you didn't ask for - some of which probably has to be done in order to make society livable for you at all - but they're doing more of it than I'd ask for, if it was up to me - supposedly on my behalf.  And they aren't doing it wrong, that I know about, or hurting me in any way, that I know about.  But they're still doing more of what they do, than I'd have really asked for... though I'm not a typical dath ilani, the typical dath ilani probably feels more on median-average like there's the right amount of Keepering going on.  Though actually, by the nature of their jobs, there's got to be more of it going on than we really know a specific reason for?  So some reasons for the Keepers' existences are hidden, and maybe my own first-impression feelings are closer to average and I'm just failing to adjust for predictable updates on the hidden info if I could see it... the whole Keeper thing is probably one of the objectively weirder institutions in dath ilan from an outside viewpoint, along with the Surreptitious Head Removers, the Official Government Con Artists, and the Planetary Emergency Rehearsal Festivals.  All of which have completely logical and reasonable reasons behind them, and are still understood and acknowledged even by Civilization generally to be some of the weirder things they have talked themselves into doing."

"Though, I mean, I don't disagree with the reasons, I can see why something like the Keepers need to exist.  Very stable geniuses can extensively develop thoughts that will wreck less stable people's minds, often without them even meaning to do that.  Even pursuing Lawfulness too far can sometimes end up that way.  Human beings are not designed to work great when we push ourselves harder and harder in the direction of Lawfulness - I mean, we're not designed at all, but I doubt it's something our distant ancestors bred themselves to be able to do safely.  I imagine that Keepers are people who by nature are smart and resilient and exceptionally stable in the face of internal insult, able to tolerate weird stuff going on inside or outside their own heads, and what they spend that internal resilience on is going way further in the direction of Law than their ancestors a million years earlier were pseudo-designed to do."

"And, I mean, I'm sure the Keepers have got a pretty good idea about who can do that by age ten.  But that's not because I could take one glance at a ten-year-old and figure out who'd be a good Keeper.  It's because, I confidently predict, the Keepers observe a lot of facts about ten-year-olds, and they keep excellent records of long-term outcomes, and they train people with very high measured intelligence to make good predictions about it.  Come to think, I wouldn't be surprised if the Keepers had a secret prediction market about me somewhere in their systems, saying exactly what my chances were of succeeding at my life goals, and people like me aren't told those predictions because that's exactly the kind of information that - can be a bit - more Lawfulness than we're really happy having in our lives.  And if you can predict that actually a kid is going to be totally okay with knowing that information, then maybe you try to make them a Keeper.  Or maybe what they predict isn't so much kids starting out imperturbable, as that you'll end up driven to face down whatever kind of internal bumps you face, in order to be able to face any kind of disturbing truth and not allow your potential to be limited by the disorder of your own mind... I don't know.  I didn't want to be a Keeper.  They weren't the kind of weird I wanted to be."

lintamande: She nods. "And - if smartness is part of it - then probably our world just doesn't have people smart enough to be Keepers, yet?"

Keltham: "I genuinely couldn't guess how much you need Keepers, and if maybe you should slap some intelligence headbands on your top geniuses and have them do their best with whatever Law they can reconstruct from what I remember.  Maybe your society does really badly without that, and does better with some Keepers that are the best Keepers you can make.  Dath ilan bred itself for intelligence over time, they didn't always have people as smart as the smartest people now, and there must also have been a time when the Keepers had much less knowledge of Law and had just started out being Keepers.  Or maybe you can get by on having a couple of advisors like Lrilatha, or building some kind of interplanar communicator that you can use to talk to - axiomites, Lrilatha called them, though she didn't think they could live here, and I'm not sure if they could do the things that a Keeper could do.  Look, I think this question is in an important sense premature?  Let's get some Chelish geniuses thinking and talking in terms that don't sound like total nonsense first, and if anybody's really good at it maybe they'll start up the Keepers here."

lintamande: The girls are so ready to get back to work on that.

Carissa Sevar: (It seems odd, to think that the Keepers would be motivated by Good, by wanting to help people. It seems like you could run something like that off the pure, selfish want to be more perfect, more like a god, held to the standards of gods - surely that's a drive, in most people, strong enough to matter far, far more than the question of whether being like that benefits other people.)

Keltham: Keltham admires (and is quietly starting to feel a bit concerned about) their apparently infinite well of drive and enthusiasm, but he needs to eat lunch and frankly allow his brain to cool down a step before resuming the Golarion Industrialization Project.  He's not going to stop them from talking about it with each other, but he needs to not talk about that during lunch, and would like something like ten minutes to himself before he talks about anything at all.

Irori: Irori has not been spending nearly enough energy to decode the actual words being exchanged on the material plane, but He has continued looking in this direction, whose spiritual position and velocity is looking increasingly relevant to His interests.

Would you look at that, somebody from this benighted corner of reality is thinking, in a surprisingly non-clueless direction, about what it even means to be a god, that isn't about touching that damnable Starstone.

Why this quite interesting event is happening inside Cheliax... is not something Irori can deduce from the information he has, but He should perhaps do a little about it. 

Irori: Irori sends a brief information packet to Asmodeus, requesting conversation under certain terms and conditions.

Asmodeus: Asmodeus does not get a lot of requests from other gods that he commit to non-intervention on the information They would like to bring to His attention for a negotiation. If it happens twice in 28800 time units then there is a single underlying cause.

Asmodeus suspects He may already know what Irori is asking about. He does not disclose this.

He agrees to the conversation.

Irori: Irori sends a potential contract to Asmodeus, regarding the treatment of a certain mortal, identity to be revealed after contract is signed.

A current mortal inhabitant of Cheliax seems to have set one foot upon the Way.

This mortal is not to be particularly hindered by Asmodeus or His deliberately dispatched agents.

If the mortal continues upon the Way, their steps shall no doubt take them beyond Cheliax in due time.  No devil shall accept sale of their soul, as Cheliax sometimes demands of its people before allowing them to leave.  Should such an event occur in Cheliax's due civil processes, any such devil is to instruct Cheliax that this mortal's soul may not be bought, but that the mortal is to be allowed to leave Cheliax regardless and not hindered in going where their footsteps take them.

The mortal's current teachers shall not be killed by Asmodeus or His dispatched agents for a period of at least one year.  If Asmodeus wants to wipe out their teachers after that, Irori shall not interfere.  (His Way is not a Way of rendering a mortal's path easy, and such events have not uncommonly spurred others on their Way.)

For this boon, Irori offers a relatively small amount of energy in payment; the request doesn't call on Asmodeus to make any urgent, costly revelations.  If the mortal falls before Cheliax's ordinary challenges, then so be it.  Indeed, Irori is offering an energy payment barely more than the cost to Asmodeus to thus instruct His relevant soul-buying devils to refuse a certain contract.  Irori would offer a greater fraction of the gains from trade, if not for Asmodeus's barely-Lawful tendency to selectively accept contracts on which He can screw others over, a tendency which Irori needs to take into account when offering prices.

Asmodeus: That squirrel. Everybody sure does love that squirrel. 

It is a cheaper ask than Abadar's, and in fact wholly encompassed by it already; He's already not allowed to keep that squirrel's soul. Which Irori does not know, because this is the fun kind of negotiation that isn't occurring with mutual access to all relevant information.

He observes that Cheliax has departing persons sell their soul for reasons, mainly that it keeps them from endangering Cheliax in ways that require costly intervention, and that the cost to Him of a troublemaker running around exceed in expectation the cost of informing His devils not to take the contract. And that the sort of soul that might find Irori's Way is an unusually valuable one to Him, too. The price isn't high enough. Irori would probably abandon the negotiation, at this point, if the squirrel were only a normal amount of promising; but Asmodeus has the secret information that this is in fact an EXCEPTIONALLY BIZARRE squirrel and so He predicts Irori is, actually, willing to pay more. 

Irori: Irori will go as high as the expected cost to a god of a mortal troublemaker, as well as the expected cost of informing His devils in due time, plus a bare margin of profit to make the contract beneficial to Asmodeus at all.  Asmodeus will wantonly wreck this soul for no good reason in the afterlife, if Asmodeus gets it, so Irori does not accept that argument.

It would be a higher price, but Irori needs to take into account that Asmodeus is much more likely to accept this contract if it is in some way cheaper to Asmodeus or less beneficial to Irori than expected.  That is as high as Irori is willing to go; if Asmodeus wants different prices, He needs to become a different kind of being.  Irori would be overjoyed to explain the relevant changes if Asmodeus is interested in correcting His flaws.

Asmodeus: Deal. Pleasure doing business. What squirrel.

Irori: Irori transmits the identifying info for one Carissa Sevar, and goes about His Way.

Asmodeus: Seriously? That one? 

Keltham: So what kind of group lunch facilities do they have around here?

lintamande: The grand dining hall has a spread of various foods; it is very abundant, by Golarion standards. The girls are mostly eating and sometimes speaking quietly to each other.

lintamande: It is not traditional at Chelish schools to talk much about your classes. After all, your classmates are your competitors, not your friends; sometimes it is mutually beneficial to collaborate on some problems, because two people are smarter than one, but it'd be stupid and pathetic, to try to build friendships out of that. 

These classes are hard not to talk about though. They are compromising by talking about their teacher. This constrains them to things it's fine if he overhears, or to speaking Infernal, which he might think is odd; some of them have bland conversations with innuendo that should be hard for him to catch, and some go for Infernal and wait to see if they'll get slapped for it. 

Conversations physically more distant from Keltham get steadily more interesting.

"I think it must be taught very differently in Hell because most people couldn't learn this way at all."

"I think Hell is doing - something different - shaping the way your intuition-brain works, instead of teaching you how to override it with - formal precision - there are devils that don't have high intelligence, and they still have it -"

"And shaping the intuitions requires suffering because it's the - language our subconscious speaks. But teaching how to override it with formal precision doesn't necessarily -"

"We're still going to have to get the intuitions later, though."

Keltham: Keltham quietly eats his food for at least the first ten minutes.  Insofar as his brain isn't just plain resting, it's going back through what he said to check whether he said anything spectacularly stupid.

He notes, absently, some Chelish girls having conversations in a language he doesn't speak.  It doesn't seem particularly worrisome by comparison with people in Chelish Governance wearing intelligence headbands having conversations where Keltham can't see them at all.  His research harem is probably just discussing strategies for seducing him or something.

...should he briefly rapidly cover genetics and deliberate heritage-optimization next?  It's got a long time lag before it's useful, but that might be all the more reason for Cheliax to get started on it early.